Bah
Koordinatni sustav u ravnini
<p> (a) Riješi sustav jednadžbi:</p>
$$
\begin{cases}
3x + 2y - z = §§V0(10,20,1)§§ \
2x - y + 3z = §§V1(5,10,1)§§ \
x + 3y - 2z = §§V2(3,5,1)§§ \
\end{cases}
$$
<p> (b) Izračunaj odvod funkcije f(x)=sin(§§V3(2,4,0.5)x)§§+e
x
:</p>
$$
f'(x) = \ldots
$$
<p> (c) Riješi diferencijalnu jednadžbu:</p>
$$
\frac{dy}{dx} = §§V5(3,-2,0.5)x^2§§ - §§V6(2,-1,0.5)x§§ + §§V7(1,2,0.5)§§
$$
<p> (d) Izračunaj određeni integral:</p>
$$
\int_{§§V10(0,1,0.1)§§}^{§§V11(1,2,0.1)§§} §§V12(1,3,0.1)x^2§§ \cos(x) , dx
$$
<p> (e) Razloži funkciju u parcijalne razlomke:</p>
$$
f(x) = \frac{1}{(x - §§V13(1,2,0.1)§§)(x + §§V14(-2,1,0.1)§§)}
$$
<p> (f) Izračunaj Laplaceovu transformaciju funkcije f(t)=§§V15(3,6,1)e
§§V16(2,4,0.5)t
§§sin(§§V17(4,8,0.5)t)§§:</p>
$$
F(s) = \ldots
$$
<p> (g) Izračunaj determinantu matrice:</p>
$$
\det \begin{pmatrix}
§§V19(2,4,1)§§ & §§V20(1,3,1)§§ & §§V21(3,5,1)§§ \
0 & §§V22(4,6,1)§§ & §§V23(2,4,1)§§ \
-1 & §§V24(2,4,1)§§ & §§V25(1,3,1)§§ \
\end{pmatrix}
$$
<p> (h) Pokaži da je funkcija f(x)=x
3
invertibilna na skupu realnih brojeva.</p>
<p> (i) Riješi integralnu jednadžbu:</p>
$$
y(x) = 1 + \int_{§§V26(0,1,0.1)§§}^{x} (x-t)y(t) , dt
$$
<p> (j) Izračunaj vrijednost kompleksnog broja:</p>
$$
\frac{(1 + §§V27(2,4,0.1)i)(§§V28(3,6,1) - §§V29(4,8,0.1)i)}{§§V30(2,4,1) - §§V31(3,6,1)i}
<p> (b) Izračunaj kompleksne korijene jednadžbe:</p>
$$
z^4 + §§V39(2,4,1) z^2 + §§V40(1,2,1) = 0
$$
<p> (c) Riješi sustav linearnih diferencijalnih jednadžbi:</p>
$$
\begin{cases}
\frac{dx}{dt} = §§V41(2,4,1)x + §§V42(1,2,1)y \
\frac{dy}{dt} = §§V43(3,6,1)x + §§V44(2,4,1)y \
\end{cases} \quad \text{s početnim uvjetima: } x(§§V45(0,1,0.1)) = §§V46(2,3,0.1), \quad y(§§V47(0,1,0.1)) = §§V48(1,2,0.1)
$$
<p> (d) Izračunaj Fourierovu transformaciju funkcije f(t)=§§V49(3,6,1)e
§§V50(2,4,0.5)t
§§sin(§§V51(4,8,0.5)t)§§:</p>
$$
F(\omega) = \ldots
$$
<p> (e) Izračunaj gradijent funkcije f(x,y,z)=§§V52(2,4,1)x
2
+§§V53(1,2,1)y
2
+§§V54(3,6,1)z
2
:</p>
$$
\nabla f(x, y, z) = \ldots
$$
<p> (f) Riješi linearnu regresiju za dane podatke:</p>
$$
\begin{vmatrix}{c|c}
x & y \
\hline
1 & §§V55(2,4,0.1) \
2 & §§V56(3,6,0.1) \
3 & §§V57(4,8,0.1) \
4 & §§V58(5,10,0.1) \
\end{vmatrix}
$$
<p> (g) Izračunaj Laplaceovu inverznu transformaciju funkcije F(s)=§§V59(1,2,1)e
§§V60(2,4,0.5)s
§§:</p>
$$
<h2>Drugi dio:</h2>
<p> (a) Riješi diferencijalnu jednadžbu:</p>